Subgroups — C₃×#10602556 #14301532

Order 1920.

888 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₃×#66132	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3965	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65504	C₂	1	2	yes	maximal, complemented
5	C₃×#64743	C₂	1	2	yes	maximal, complemented
6	#10602556	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×D₄	—	2	2	no	—
8	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
10	C₃×C₅×#127	—	2	2	no	—
11	C₃×C₅×#127	C₄	1	4	yes	complemented
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₂²⋊Dic₅	—	2	2	no	—
14	C₃×D₄×Dic₅	—	2	4	no	—
15	C₃×#4072	C₂²	1	2	yes	—
16	C₂×C₃×#681	C₂²	1	2	yes	—
17	C₃×#3965	—	2	2	no	—
18	C₃×#3964	—	2	4	no	—
19	C₃×#1887	—	5	1	no	maximal, Hall
20	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic
21	#66132	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×#3965	C₂×C₃	1	1	yes	characteristic, complemented
23	#65504	C₂×C₃	1	2	yes	complemented
24	#64743	C₂×C₃	1	2	yes	complemented
25	C₂⁴×C₃×C₅	—	2	1	no	—
26	C₂⁴×C₃×C₅	D₄	1	2	yes	—
27	C₂×C₃×C₅×D₄	—	2	8	no	—
28	C₂×C₃×C₅×D₄	—	2	2	no	—
29	C₂×C₃×C₅×D₄	—	2	2	no	—
30	C₂×C₃×C₅×D₄	—	2	1	no	—
31	C₂×C₃×C₅×D₄	—	2	2	no	—
32	C₂×C₃×C₅×D₄	—	2	2	no	—
33	C₂×C₃×C₅×D₄	—	4	8	no	—
34	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	complemented
36	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
37	C₂²×C₃×C₄×C₅	—	2	1	no	—
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂²×C₃×Dic₅	—	2	4	no	—
40	C₃×C₄²×C₅	—	2	1	no	—
41	C₃×C₄²×C₅	C₂×C₄	1	2	yes	complemented
42	C₃×C₂²⋊Dic₅	—	2	4	no	—
43	C₃×C₂²⋊Dic₅	—	2	2	no	—
44	C₃×C₄⋊Dic₅	D₄	1	4	yes	complemented
45	C₃×C₄×Dic₅	—	2	2	no	—
46	C₃×#681	—	2	2	no	—
47	C₃×#681	—	2	2	no	—
48	C₂×C₃×#168	—	2	2	no	—
49	C₂×C₃×#127	D₅	1	1	yes	characteristic
50	C₃×#518	—	5	1	no	—
51	C₃×#418	—	5	2	no	—
52	C₂×C₃×#100	—	5	1	no	—
53	C₃×#339	—	5	2	no	—
54	C₂²×C₅×D₄	—	2	2	no	—
55	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
56	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
57	C₅×#127	—	2	2	no	—
58	C₅×#127	C₃×C₄	1	4	yes	complemented
59	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₂²⋊Dic₅	—	2	2	no	—
61	D₄×Dic₅	—	2	4	no	—
62	#4072	C₂²×C₃	1	2	yes	—
63	C₂×#681	C₂²×C₃	1	2	yes	—
64	#3965	—	2	2	no	—
65	#3964	—	2	4	no	—
66	#1887	—	5	1	no	Sylow
67	C₂³×C₃×C₅	—	2	2	no	—
68	C₂³×C₃×C₅	—	2	2	no	—
69	C₂³×C₃×C₅	—	2	2	no	—
70	C₂³×C₃×C₅	—	4	4	no	—
71	C₂³×C₃×C₅	—	4	4	no	—
72	C₂³×C₃×C₅	—	4	1	no	—
73	C₂³×C₃×C₅	—	4	1	no	—
74	C₂³×C₃×C₅	C₂²⋊C₄	1	4	yes	—
75	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
76	C₃×C₅×D₄	—	2	4	no	—
77	C₃×C₅×D₄	—	2	4	no	—
78	C₃×C₅×D₄	—	2	4	no	—
79	C₃×C₅×D₄	—	4	16	no	—
80	C₃×C₅×D₄	—	4	2	no	—
81	C₂×C₃×C₄×C₅	—	2	4	no	—
82	C₂×C₃×C₄×C₅	—	2	2	no	—
83	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	#41	1	2	yes	—
87	C₂×C₃×Dic₅	—	2	4	no	—
88	C₂×C₃×Dic₅	—	8	4	no	—
89	C₃×#168	—	4	2	no	—
90	C₂²×C₃×D₄	—	2	2	no	—
91	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
92	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
93	C₂×C₃×M₁₆	—	5	2	no	—
94	C₃×#127	—	2	2	no	—
95	C₃×#127	Dic₅	1	4	yes	complemented
96	C₃×C₄×D₄	—	10	4	no	—
97	C₃×#117	—	5	2	no	—
98	C₂×C₃×C₂²⋊C₄	—	10	2	no	—
99	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
100	C₃×#102	—	10	4	no	—
101	C₃×#100	—	10	2	no	—
102	C₂⁴×C₅	—	2	1	no	—
103	C₂⁴×C₅	C₃×D₄	1	2	yes	—
104	C₂×C₅×D₄	—	2	8	no	—
105	C₂×C₅×D₄	—	2	2	no	—
106	C₂×C₅×D₄	—	2	2	no	—
107	C₂×C₅×D₄	—	2	1	no	—
108	C₂×C₅×D₄	—	2	2	no	—
109	C₂×C₅×D₄	—	2	2	no	—
110	C₂×C₅×D₄	—	4	8	no	—
111	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
112	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	complemented
113	C₂×C₅×D₄	C₃×D₄	1	2	yes	—
114	C₂²×C₄×C₅	—	2	1	no	—
115	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
116	C₂²×Dic₅	—	2	4	no	—
117	C₄²×C₅	—	2	1	no	—
118	C₄²×C₅	C₂×C₃×C₄	1	2	yes	complemented
119	C₂²⋊Dic₅	—	2	4	no	—
120	C₂²⋊Dic₅	—	2	2	no	—
121	C₄⋊Dic₅	C₃×D₄	1	4	yes	complemented
122	C₄×Dic₅	—	2	2	no	—
123	#681	—	2	2	no	—
124	#681	—	2	2	no	—
125	C₂×#168	—	2	2	no	—
126	C₂×#127	C₃×D₅	1	1	yes	characteristic
127	#518	—	5	1	no	—
128	#418	—	5	2	no	—
129	C₂×#100	—	5	1	no	—
130	#339	—	5	2	no	—
131	C₂²×C₃×C₅	—	2	2	no	—
132	C₂²×C₃×C₅	—	2	4	no	—
133	C₂²×C₃×C₅	—	4	4	no	—
134	C₂²×C₃×C₅	—	4	4	no	—
135	C₂²×C₃×C₅	—	4	4	no	—
136	C₂²×C₃×C₅	—	4	2	no	—
137	C₂²×C₃×C₅	—	8	2	no	—
138	C₂²×C₃×C₅	—	8	2	no	—
139	C₂²×C₃×C₅	—	8	2	no	—
140	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₃×C₅	#120	1	1	yes	characteristic
142	C₂²×C₃×C₅	#123	1	1	yes	characteristic
143	C₃×C₄×C₅	—	2	4	no	—
144	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
145	C₃×C₄×C₅	#121	1	2	yes	—
146	C₃×Dic₅	—	4	4	no	—
147	C₂⁴×C₃	—	2	1	no	—
148	C₂⁴×C₃	C₂²⋊D₅	1	2	yes	—
149	C₂×C₃×D₄	—	2	8	no	—
150	C₂×C₃×D₄	—	2	2	no	—
151	C₂×C₃×D₄	—	2	2	no	—
152	C₂×C₃×D₄	—	2	2	no	—
153	C₂×C₃×D₄	—	2	2	no	—
154	C₂×C₃×D₄	—	2	1	no	—
155	C₂×C₃×D₄	—	4	8	no	—
156	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
157	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	complemented
158	C₂×C₃×D₄	C₂²⋊D₅	1	2	yes	—
159	C₂²×C₃×C₄	—	2	1	no	—
160	C₂²×C₃×C₄	—	10	4	no	—
161	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
162	C₃×M₁₆	—	10	2	no	—
163	C₃×M₁₆	—	10	2	no	—
164	C₂×C₃×C₈	—	10	2	no	—
165	C₃×C₄⋊C₄	—	5	4	no	—
166	C₃×C₂²⋊C₄	—	10	4	no	—
167	C₃×C₂²⋊C₄	—	10	2	no	—
168	C₃×C₄²	—	2	1	no	—
169	C₃×C₄²	—	10	2	no	—
170	C₃×C₄²	C₂×Dic₅	1	2	yes	complemented
171	C₂³×C₅	—	2	2	no	—
172	C₂³×C₅	—	2	2	no	—
173	C₂³×C₅	—	2	2	no	—
174	C₂³×C₅	—	4	4	no	—
175	C₂³×C₅	—	4	4	no	—
176	C₂³×C₅	—	4	1	no	—
177	C₂³×C₅	—	4	1	no	—
178	C₂³×C₅	C₃×C₂²⋊C₄	1	4	yes	—
179	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
180	C₅×D₄	—	2	4	no	—
181	C₅×D₄	—	2	4	no	—
182	C₅×D₄	—	2	4	no	—
183	C₅×D₄	—	4	16	no	—
184	C₅×D₄	—	4	2	no	—
185	C₂×C₄×C₅	—	2	4	no	—
186	C₂×C₄×C₅	—	2	2	no	—
187	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
189	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅	C₃×#41	1	2	yes	—
191	C₂×Dic₅	—	2	4	no	—
192	C₂×Dic₅	—	8	4	no	—
193	#168	—	4	2	no	—
194	C₂²×D₄	—	2	2	no	—
195	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
196	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
197	C₂×M₁₆	—	5	2	no	—
198	#127	—	2	2	no	—
199	#127	C₃×Dic₅	1	4	yes	complemented
200	C₄×D₄	—	10	4	no	—
201	#117	—	5	2	no	—
202	C₂×C₂²⋊C₄	—	10	2	no	—
203	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
204	#102	—	10	4	no	—
205	#100	—	10	2	no	—
206	C₂×C₃×C₅	—	2	1	no	—
207	C₂×C₃×C₅	—	2	1	no	—
208	C₂×C₃×C₅	—	4	4	no	—
209	C₂×C₃×C₅	—	8	2	no	—
210	C₂×C₃×C₅	#386	1	1	yes	characteristic
211	C₂×C₃×C₅	#453	1	2	yes	—
212	C₂³×C₃	—	2	2	no	—
213	C₂³×C₃	—	2	2	no	—
214	C₂³×C₃	—	2	2	no	—
215	C₂³×C₃	—	4	4	no	—
216	C₂³×C₃	—	4	4	no	—
217	C₂³×C₃	—	4	1	no	—
218	C₂³×C₃	—	4	1	no	—
219	C₂³×C₃	C₂²⋊Dic₅	1	4	yes	—
220	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
221	C₃×D₄	—	2	4	no	—
222	C₃×D₄	—	2	4	no	—
223	C₃×D₄	—	2	4	no	—
224	C₃×D₄	—	4	16	no	—
225	C₃×D₄	—	4	2	no	—
226	C₂×C₃×C₄	—	2	4	no	—
227	C₂×C₃×C₄	—	2	2	no	—
228	C₂×C₃×C₄	—	10	4	no	—
229	C₂×C₃×C₄	—	40	4	no	—
230	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
231	C₂×C₃×C₄	#711	1	2	yes	—
232	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
234	C₃×C₈	—	20	2	no	—
235	C₂²×C₅	—	2	2	no	—
236	C₂²×C₅	—	2	4	no	—
237	C₂²×C₅	—	4	4	no	—
238	C₂²×C₅	—	4	4	no	—
239	C₂²×C₅	—	4	4	no	—
240	C₂²×C₅	—	4	2	no	—
241	C₂²×C₅	—	8	2	no	—
242	C₂²×C₅	—	8	2	no	—
243	C₂²×C₅	—	8	2	no	—
244	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
245	C₂²×C₅	C₃×#120	1	1	yes	characteristic
246	C₂²×C₅	C₃×#123	1	1	yes	characteristic
247	C₄×C₅	—	2	4	no	—
248	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
249	C₄×C₅	C₃×#121	1	2	yes	—
250	Dic₅	—	4	4	no	—
251	C₂⁴	—	2	1	no	—
252	C₂⁴	C₃×C₂²⋊D₅	1	2	yes	—
253	C₂×D₄	—	2	8	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	4	8	no	—
260	C₂×D₄	C₂×C₃×Dic₅	1	2	yes	—
261	C₂×D₄	C₂×C₃×Dic₅	1	2	yes	complemented
262	C₂×D₄	C₃×C₂²⋊D₅	1	2	yes	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	10	4	no	—
265	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
266	M₁₆	—	10	2	no	—
267	M₁₆	—	10	2	no	—
268	C₂×C₈	—	10	2	no	—
269	C₄⋊C₄	—	5	4	no	—
270	C₂²⋊C₄	—	10	4	no	—
271	C₂²⋊C₄	—	10	2	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	10	2	no	—
274	C₄²	C₂×C₃×Dic₅	1	2	yes	complemented
275	C₃×C₅	#1887	1	1	yes	characteristic, complemented, Hall
276	C₂²×C₃	—	2	2	no	—
277	C₂²×C₃	—	2	4	no	—
278	C₂²×C₃	—	4	4	no	—
279	C₂²×C₃	—	4	4	no	—
280	C₂²×C₃	—	4	4	no	—
281	C₂²×C₃	—	4	2	no	—
282	C₂²×C₃	—	8	2	no	—
283	C₂²×C₃	—	8	2	no	—
284	C₂²×C₃	—	8	2	no	—
285	C₂²×C₃	#4081	1	1	yes	characteristic
286	C₂²×C₃	#4083	1	1	yes	characteristic
287	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
288	C₃×C₄	—	2	4	no	—
289	C₃×C₄	—	20	4	no	—
290	C₃×C₄	D₄×Dic₅	1	2	yes	—
291	C₃×C₄	#4085	1	2	yes	—
292	C₂×C₅	—	2	1	no	—
293	C₂×C₅	—	2	1	no	—
294	C₂×C₅	—	4	4	no	—
295	C₂×C₅	—	8	2	no	—
296	C₂×C₅	C₃×#386	1	1	yes	characteristic
297	C₂×C₅	C₃×#453	1	2	yes	—
298	C₂³	—	2	2	no	—
299	C₂³	—	2	2	no	—
300	C₂³	—	2	2	no	—
301	C₂³	—	4	4	no	—
302	C₂³	—	4	4	no	—
303	C₂³	—	4	1	no	—
304	C₂³	—	4	1	no	—
305	C₂³	C₃×C₂²⋊Dic₅	1	4	yes	—
306	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
307	D₄	—	2	4	no	—
308	D₄	—	2	4	no	—
309	D₄	—	2	4	no	—
310	D₄	—	4	16	no	—
311	D₄	—	4	2	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	10	4	no	—
315	C₂×C₄	—	40	4	no	—
316	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
317	C₂×C₄	C₃×#711	1	2	yes	—
318	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
319	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
320	C₈	—	20	2	no	—
321	C₂×C₃	—	2	1	no	—
322	C₂×C₃	—	2	1	no	—
323	C₂×C₃	—	4	4	no	—
324	C₂×C₃	—	8	2	no	—
325	C₂×C₃	#65366	1	2	yes	—
326	C₂×C₃	#65509	1	1	yes	characteristic
327	C₅	C₃×#1887	1	1	yes	characteristic, complemented, Sylow
328	C₂²	—	2	2	no	—
329	C₂²	—	2	4	no	—
330	C₂²	—	4	4	no	—
331	C₂²	—	4	4	no	—
332	C₂²	—	4	4	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	C₃×#4081	1	1	yes	characteristic
338	C₂²	C₃×#4083	1	1	yes	characteristic
339	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
340	C₄	—	2	4	no	—
341	C₄	—	20	4	no	—
342	C₄	C₃×D₄×Dic₅	1	2	yes	—
343	C₄	C₃×#4085	1	2	yes	—
344	C₃	#10602556	1	1	yes	characteristic, direct product factor, Sylow
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	4	4	no	—
348	C₂	—	8	2	no	—
349	C₂	C₃×#65366	1	2	yes	—
350	C₂	C₃×#65509	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#4083
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting, prime radical	C₂×C₃×C₅×#127	C₂
socle	C₂²×C₃×C₅	#120
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×#127	D₅
hypocenter	C₅	C₃×#1887
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#127

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃×#66132	C₂	semidirect
3. C₂×C₃×#3965	C₂	semidirect
4. C₃×#65504	C₂	semidirect
5. C₃×#64743	C₂	semidirect
6. #10602556	C₃	direct (N × H)
11. C₃×C₅×#127	C₄	semidirect
21. #66132	C₂×C₃	semidirect
22. C₂×#3965	C₂×C₃	semidirect
23. #65504	C₂×C₃	semidirect
24. #64743	C₂×C₃	semidirect
35. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
41. C₃×C₄²×C₅	C₂×C₄	semidirect
44. C₃×C₄⋊Dic₅	D₄	semidirect
58. C₅×#127	C₃×C₄	semidirect
95. C₃×#127	Dic₅	semidirect
112. C₂×C₅×D₄	C₂×C₃×C₄	semidirect
118. C₄²×C₅	C₂×C₃×C₄	semidirect
121. C₄⋊Dic₅	C₃×D₄	semidirect
157. C₂×C₃×D₄	C₂×Dic₅	semidirect
170. C₃×C₄²	C₂×Dic₅	semidirect
199. #127	C₃×Dic₅	semidirect
261. C₂×D₄	C₂×C₃×Dic₅	semidirect
274. C₄²	C₂×C₃×Dic₅	semidirect
275. C₃×C₅	#1887	semidirect
327. C₅	C₃×#1887	semidirect
344. C₃	#10602556	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×C₅×#127	2
C₃×#66132	2
C₂×C₃×#3965	2
C₃×#65504	2
C₃×#64743	2
#10602556	3
C₃×#1887	5

Subgroups — C3×#10602556 #14301532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602556 #14301532